# The quantum information manifold for epsilon-bounded forms

by M. Grasselli and R. F. Streater

## ABSTRACT

Let H be a self-adjoint operator greater than I and let V be a form-small perturbation such that the operator-norm of R^{1/2+\epsilon}VR^{1/2-\epsilon} is finite. Here, epsilon lies in (0,1) and R is the inverse of H. Suppose that there exists a positive beta less than 1, such that Z, the trace of exp(-beta H), is finite. Let Z(V) be the trace of exp(-beta(H+V)). Then we show that the free energy psi:=log Z is a real analytic function of V in the sense of Fréchet, and that the family of density operators defined in this way is an analytic manifold.

This is available from the archives, math-ph/9910031.

The paper appears as Reports on Mathematical Physics, 46,325-335, 2000. It improves on the result in The Analytic Quantum Information Manifold because the perturbations allowed in the present paper include some form-bounded perturbations of the Hamiltonian that are not operator bounded. In such cases the free energy is still analytic in the parameter measuring the perturbation, but in a smaller circle than if the perturbation is operator bounded.